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JEE Advanced2015MathematicsThree Dimensional GeometryActual

In R 3 , let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P 1 : x + 2 y - z + 1 = 0 and P 2 : 2 x - y + z - 1 = 0 . Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P 1 . Which of the following points lie (s) on M ?

Options

  1. A0 , - 5 6 , - 2 3
  2. B- 1 6 , - 1 3 , 1 6
  3. C- 5 6 , 0 , 1 6
  4. D- 1 3 , 0 , 2 3

Correct answer

A. 0 , - 5 6 , - 2 3

Step-by-step solution

Line L will lie on one of the angle bisector planes and will be parallel to line of intersection of the given planes. Also locus of foot of perpendicular of line L in plane P 1 will be line M, which will be parallel L. Foot of perpendicular from (0, 0, 0) on plane P 1 is - 1 6 , - 1 3 , 1 6 Hence equation of line M x + 1 6 1 = y + 1 3 - 3 = z - 1 6 - 5 Where i ^ - 3 j ^ - 5 k ^ is vector parallel to line of intersection of P 1 and P 2 . On checking 0 , - 5 6 - 2 3 , - 1 6 , - 1 3 , 1 6 satisfy the above equation.

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